Principle of Least Action

The least-action principle is an assertion about the
nature of motion that provides an alternative approach to mechanics
completely independent of Newton's laws. Not only does the least-action
principle offer a means of formulating classical mechanics that is
more flexible and powerful than Newtonian mechanics, [but also] variations
on the least-action principle have proved useful in general relativity
theory, quantum field theory, and particle physics. As a result,
this principle lies at the core of much of contemporary theoretical
physics.
Thomas A. Moore "Least-Action Principle" in Macmillan Encyclopedia
of Physics, John Rigden, editor, Simon & Schuster Macmillan, 1996, Volume
2, page 840.

The principle of least action (more correctly, the principle of
stationary action) has wide applicability in undergraduate physics
education, from mechanics in introductory classes through electricity
and magnetism, quantum mechanics, special and general relativity—and
it provides a deep foundation for advanced subjects and current
research.

Interactive SoftwarePrinciple of Least Action Interactive (zip
archive of all files or on-line
JAVA applications) by Slavomir Tuleja and Edwin F. Taylor.
An interactive introduction to the Principle of Least Action. Twenty-six
questions for the student to answer using five different JAVA interactive
displays. If you are having trouble running the JAVA applets, read
the readme file (pdf
format).

The software ActionClockTicks by Slavo Tuleja uses the action
principle to construct trajectories between fixed initial and final
points for projectile motion, satellite motion, simple harmonic
oscillator, and a moon shot. The total time for the trajectory
is set by the user. Play this program online or download it to
your computer. The open source JAVA code of the program is also
available.

Action Summary by Jozef Hanc, Jon Ogborn, Edwin Taylor,
with software by Slavomir Tuleja. A conference proceeding that
summarizes action, starting with dramatized appearances of the
originators from Fermat and Maupertuis to Hamilton and Feynman.
Subsequent papers introduce action formalism, apply it to teaching
introductory quantum mechanics, and trace the scope of applications
of action from quantum field theory to cosmology.
[ Download PDF file (~1.9MB) ] [ Back to overview ]

"A Call to Action," Edwin F. Taylor. Guest Editorial, American Journal of
Physics,Vol. 71, No. 5, May 2003, pages 423-425. Outlines the case for using the principle
of least action in undergraduate physics classes.
[ Read on-line as HTML ] [ Download
PDF file (~39K) ] [ Back to overview ]
"Deriving the nonrelativistic principle of least action from the Schwarzschild
metric and the Principle of Maximal Aging" Edwin F. Taylor. Unpublished appendix
to "A Call to Action." In general relativity a particle moves along the worldline
of maximal proper time (maximal aging). In the limit of small spacetime curvature
and low velocity this reduces to the principle of least action, as shown
in this paper for a free particle external to a spherically symmetric, nonrotating
center of gravitational attraction.
[ Download PDF file (~138K) ] [ Back
to overview ]
"Action: Forcing Energy to Predict Motion," Dwight E. Neuenschwander, Edwin F.
Taylor, and Slavomir Tuleja, The Physics Teacher, Vol. 44, March 2006, pages
146-152. Scalar energy is employed to predict motion instead of the vector Newton's
law of motion.
[ Download PDF file (~260K) ]
[ Back
to overview ]"Variational mechanics in one and two dimensions" Jozef Hanc, Edwin F. Taylor,
Slavomir Tuleja. American Journal of Physics, Vol. 73, No. 7, July 2005, pages
603-610. Heuristic derivations of the Euler-Maupertuis abbreviated action and
the Hamilton action.
[ Read on-line as HTML ] [ Download
PDF file (~110K) ] [ Back to overview ]"Symmetries and conservation laws: Consequences of Noether's theorem," Jozef
Hanc, Slavomir Tuleja, and Martina Hancova, American Journal of Physics, Vol.
72, No. 4, April 2004, pages 428-435. Derives conservation laws from symmetry
operations using the principle of least action. These derivations are examples
of Noether's Theorem.
[ Read on-line as HTML ] [ Download
PDF file (~337K) ] [ Back to overview ]"Deriving Lagrange's equations using elementary calculus," Jozef Hanc, Edwin F. Taylor, and Slavomir Tuleja. American
Journal of Physics, Vol. 72, No. 4, April 2004, pages 510-513. Lagrange's equations, alternatives to F=ma, are usually derived from the principle of least action using the calculus of variations. This paper derives them using elementary calculus.
[ Read on-line as HTML ] [ Download
PDF file (~83K) ] [ Back to overview ]"Simple derivation of Newtonian mechanics from the principle of least action," Jozef
Hanc, Slavomir Tuleja, Martina Hancova. American Journal of Physics, Vol. 71.
No. 4, April 2003, pages 386 - 391. Derives Newton's laws of motion from the
principle of least action.
[ Read on-line as HTML ] [ Download
PDF file (~250K) ] [ Back to overview ]"Quantum physics explains Newton's laws of motion," Jon Ogborn and Edwin F. Taylor, Physics
Education, Vol. 40, No. 1, 2005, pages 26-34. Richard Feynman expressed the quantum mechanics of particle motion in the command, "Explore all paths." In the limit of large mass, this command goes over into Newton's law of motion and the principle of least action.
[ Download PDF file (~140K) ] [ Back
to overview ]"From
conservation of energy to the principle of least action: A story line," Jozef
Hanc and Edwin F. Taylor, American Journal of Physics, Vol. 72, No.
4, Aril 2004, pages 514-521. Conservation of energy is sufficient to predict
motion in one
dimension and for systems whose motion can be expressed as one independent coordinate.
This prediction can also be used to introduce the principle of least action and
Lagrange's equations.
[ Read on-line as HTML ] [ Download
PDF file (~200K) ] [ Back to overview ]"Getting the Most Action from the Least Action: A proposal," Thomas A. Moore, American
Journal of Physics, Vol. 72, No. 4, April 2004, pages 522-527. The principle of least action is a powerful addition to upper undergraduate courses for physics majors, modifying the selection of topics and presenting advanced topics in a more contemporary way.
[ Read on-line as HTML ] [ Download
PDF file (~87K) ] [ Back to overview ]
"When action is not least," C. G. Gray and Edwin F. Taylor, American
Journal of Physics, Vol. 75, No. 5, May 2007, pages 434-458. Action
is a minimum along a sufficiently short worldline in all potentials
and along worldlines of any length in some potentials. For long
enough worldlines in a majority potentials, however, the action
is a saddle point, that is, a minimum with respect to some nearby
alternative curves and a maximum with respect to others. The action
is never a true maximum.
[ Download PDF file (~1.0MB)
] [ Back
to overview ]